# Linear representation theory of groups of order 120

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 120.
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## The three non-solvable groups

There are three non-solvable groups of order 120, whose linear representation theory deserves special mention:

Group Quasisimple group? Almost simple group? Linear representation theory
symmetric group:S5 No Yes linear representation theory of symmetric group:S5
special linear group:SL(2,5) Yes No linear representation theory of special linear group:SL(2,5)
direct product of A5 and Z2 No No linear representation theory of direct product of A5 and Z2

## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization
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